Joint Model for Exchange Rate Dynamics and Influence of ...aldous/Research/Ugrad/Jian_Wang.pdf · Influence of Presetting Correlation between Stock Price and Exchange Market ... dynamic

Fall 2014—Honor Thesis--Statistics

Joint Model for Exchange Rate Dynamics and

Influence of Presetting Correlation between Stock

Price and Exchange Market

Jian Wang

University of California, Berkeley

[email protected]

I. INTRODUCTION:

American Depository Receipts (ADRs) are securities issued by non-US companies to cross list

their stocks on both domestic and foreign stock exchanges. In addition to stocks, there are also

options traded in these markets denominated in their respective currencies. As such, ADR

options are exposed to foreign exchange rate risk. In the summer research in Columbia

University we studied the implied volatilities of options issued by a non-US company and their

US traded ADR counterparts. Under the stochastic assumption of exchange rate we applied the

joint model for implied volatility and exchange rate to explain the options price discrepancies.

For each company by mean square minimization we extracted the exchange rate implied

volatility, which turned to be pretty effective and could catch the clue of currency depreciation

up to three months in advance. In the dynamic model, the correlation between stock price and

exchange rate was a critical factor determining the exchange rate implied volatility and we

maximized the mean square error over the exchange rate volatility and correlation. Yet the

implied correlation was quite unstable and if we constrained the correlation between -1 and 1 we

actually got lots of extreme implied correlation, which was not realistic. In the independent

research this semester, we can examine the sensitivity of implied exchange rate volatility respect

to correlation. By reasonable preset of the correlation, the expression of implied volatility

becomes explicit for direct analysis. There are three different measurements of correlation,

dynamic correlation over a short period, correlation as regression line over long period and

industrialized correlation to exchange rate. Though different measurement of correlation between

foreign stock market and exchange rate caught the similar trend, they gave significant different

levels of implied exchange rate volatility. By examining the results of three measurements of

correlation, we can compare the advantages and limitations for these methods. The study

improves the model we built in the summer to extract foreign exchange rate implied volatility

and provides a procedure to better understanding the perceived exchange rate risks for different

stocks across exchanges.

mailto:[email protected]


II. JOINT MODEL FOR IMPLIED VOLATILITY AND EXCHANGE RATE

Theoretical background1:

In the whole research, for an ADR option, we deal with the foreign stock struck in domestic

currency (here USD)2. The dynamic model of the entire economy, under objective measure P, is

as follows

dX = XαX dt + XσX dW

dSd = Sd αd dt + Sd σddW

dSf = Sfαfdt + SfσfdW

dBd = rdBd dt

dBf = rfBfdt

where

W =

W 1

W 2

W 3

Is a three-dimensional Winner process (with independent components)

For the foreign call struck in domestic currency, the claim, expressed in domestic term, is given

by

Zd = max X T ∗ Sf T − K, 0

Use the Black-Schoels formula we can obtain the price function and the implied volatility is

given by

σf + σX

where σf is the foreign stock volatility and σX is the foreign exchange rate volatility

by no-arbitrage argument, the implied volatility of foreign stock struck in domestic currency

should equal to the implied volatility of domestic options σd

σf + σX = σd

1 Detailed proof refers to Bjoerk book Arbitrage Theory in Continuous Time Chapter 12

2 See Appendix 1 for all the ADRs used in this project


if we denote ρ as the correlation between σf and σX , the equation is

σf2 + σX

2 + 2σfσXρ=σd

And we can get the implied exchange rate volatility σX = σf2ρ2 − (σf

2 − σd2) - σfρ

Mean Square minimization3:

In practice, given a stock for a specific day, there are many option choices with different strike

price and expiration date. Notice the implied volatilities have a smile curve. After converting the

strike price of foreign options into domestic price, we can compare the implied volatilities of

foreign options and domestic options for the same stock. Here we use UBS as illustration4.

The data is from Bloomberg on May 28th

.The implied volatility from emerging market is

shown in red and the implied volatility from the ADR market is shown in black. The blue dot

points the current spot price of ADR.

We see the implied volatility differs a lot in some extreme strike price, which is actually not a

problem since in reality those options are not traded in the market. We only need concern about

the implied volatility of options with the strike around the spot price.

3Corroborative work with Professor Tim Leung and Connie Lee in Columbia University in summer 2014

4 See Appendix 2 for more examples and illustrations of implied volatility curves


As expected, the curve of implied volatility in foreign market is higher than that in domestic

market (U.S. market), since the ADRs in U.S. market also carry the currency risk that increase

the options price.

We aim to find ρ and σX that minimize the mean square error term between two curves

( ρ , σX )=argmin (σf2 + σX

2 + 2σfσXρ − σd2)i

ni=1

subject to −1 ≤ ρ ≤ 1; σX ≥ 0

where n is total number of points of foreign option strike price struck in USD, σfi is foreign

implied volatility for ith strike price expressed in USD and σdi is the ADR implied volatility of

the same strike price computed by interpolation.

In practice since the strike prices are not exactly the same and we need interpolate either σfi


(ADR based methods) or σdi (local based methods) or mix them weighted by the number of

points. Here we examine the implied exchange rate volatility from the implied volatility disparity

(data from 2014/01/02 to 2014/05/29 with expiration date 2014/12/15) of Deutsche Bank (DB in

ADR market and DBK in Germen market). The time series of σX gave us a general impression

of exchange rate dynamics.

Due to the constraint of σX very rarely the optimal σX is 0 and we can just interpret it as the

very low exchange rate volatility. In general the analysis gave us a good impression about the

implied exchange rate volatility, from which we can know people’s expectation about the

exchange rate in advance from the options implied volatility disparity.

The similar analysis of implied correlation between foreign stock prices and exchange rate is

comparatively less significant.


Due to the constraint of correlation in the minimization problem there is lots of extreme value

as optimal correlation especially for the ADR based methods. First the extreme value lost its

statistical significance since the optimal value of σX here may be far from correct implied

value without the constraint of correlation. In addition we should look at the nature of implied

correlation and implied exchange rate volatility. People’s expectation could be quite fluctuated in

respond to the information in exchange market, which corresponding the time series ofσX . Yet

the other factor, the implied correlation between the foreign stock and exchange rate represented

people view about the correlation, which is hard to change dramatically during a short time

period. The relative stable implied correlation and fluctuate exchange rate volatility is more

reasonable. Thus to understand the implied exchange rate dynamics we need further exploration

about the other factor, the implied correlation between foreign stock price and exchange rate.

III. SENTIVITY OF IMPLIED EXCHANGE RATE VOLATILITY TO

CORRELATION

The historical correlation between stock price and exchange rate actually is not stable. Here

we take DBK and EUR as illustration

The historical correlation fluctuate from -0.5 to 0.6, which is a very broad range of correlation.

Thus we need to examine how sensitive of to correlation.

To see how the σX is affected by rho I apply the same methods of MSE to DBK and

find the implied volatility of exchange rate but for different rhos obtained from different

time scale .

-0.5000

-0.4000

-0.3000

-0.2000

-0.1000

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

Realized Correlation

Realized Correlation


Rho=-

0.18 (1w)

-

0.62

(2w)

-

0.52

(1m)

-

0.55

(2m)

-

0.32

(3m)

-

0.14

(4m)

-

0.06

(6m)

-

0.005

(1y)

0.3

0

(2y)

2014.07.18 13.11 28.01 24.32 25.29 17.56 12.14 10.16 8.91 4.61

2014.10.17 15.74 30.62 26.91 27.88 20.16 14.77 12.77 11.48 6.64

2015.01.16 16.56 31.41 27.71 28.67 20.97 15.59 13.58 12.28 7.31

2016.01.15 18.50 33.88 30.04 31.04 23.07 17.49 15.40 14.03 8.66

𝜎𝑋 from empirical data for last two years:

9.47

The σX is actually quite sensitive to rho so we need choose the proper rho carefully.

One possible solution is to use the rho from same length of maturity (i.e. if we

calculate σX of options with three month maturity, we use the rho from past three

months)





IV. DIFFERENT MEASUREMENTS OF CORRELATION

Instead of minimizing the mean square error over σX and ρ we can preset the

correlation and calculate the implied exchange rate volatility. In the original methods,

the implied correlation is extremely unstable and as the analysis above it could

significantly influence the optimal exchange rate volatility. By presetting the correlation,

the minimization problem is simplified to the quadratic minimization

σX =argmin (σf2 + σX

2 + 2σfσXρ − σd2)i

ni=1


We can give the explicit formula for σX = σfρ

ni=1

n

From the explicit formula the implied exchange rate volatility is determined by the foreign

implied volatility and correlation between foreign stock and exchange market. The fluctuated

foreign stock and s strong positive correlation between the stock and exchange market

correspond the larger implied exchange rate volatility. By presetting the correlation, the pattern

of implied exchange rate volatility becomes more clear and simple.

The other important parity also came from the explicit formula. The implied exchange rate

volatility represents people’s expectation about the volatility of exchange rate and only depends

on the information on the macroeconomic factors. In other words, the ADRs implication is the

approach to get the implied exchange volatility and theoretically using different ADR stock we

should get the same implied volatility. Back to the formula, which says σX is equal to the

average of product of stock price implied volatility and its correlation with exchange rate for

different stocks. In this project the parity is used to test the result of σX . One potential application

is to compare the implied correlation for different stocks if we know the implied exchange rate

volatility, which may be further studied in the future.

Finally we can use three different measurements of correlation

Correlation as time series over a short period

As discussed previously, we can use the realized correlation as the measurement of

implied correlation since people’s view of correlation is relatively stable.

The realized correlation from 2014.01.02 to 2014.07.18 is within the range between -

0.4 and 0.5. Compared to the implied correlation we got from minimizing both σX and ρ,

the change of realized correlation changes steadily.

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Realized Correlation (DBK)



Correlation as regression over long period

Another measurement of correlation is the average correlation over long period and

we can regard the correlation as coefficients of regression of change of exchange rate

against stock price. Although compared to the first measurement it’s not dynamic,

instead we can get the confidence interval of the correlation, which will gave us the

range of the implied exchange rate volatility.

lm(formula = diff(DBK[, 2]) ~ diff(DBK[, 1]))

Residuals:

Min 1Q Median 3Q Max

-0.0084750 -0.0018027 -0.0001058 0.0012873 0.0109077

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -1.713e-05 2.347e-04 -0.073 0.942

diff(DBK[, 1]) 8.777e-05 4.956e-04 0.177 0.860

Residual standard error: 0.002961 on 159 degrees of freedom

Multiple R-squared: 0.0001972, Adjusted R-squared: -0.006091

F-statistic: 0.03136 on 1 and 159 DF, p-value: 0.8597

The standard deviation of the slope is pretty small and thus we have a narrow

confidence interval of the correlation as shown in the dashed line. The long-run


correlation between stock price and exchange rate is approximate zero with a very tiny

range.

Industrialized Correlation for Different Time Scale

Since the implied exchange rate volatility represent the fluctuation of exchange rate

and should be independent of our choice of specific stocks, we can also looked up for the

aggregate stock market index. Here we can preset the correlation as the correlation

between the bank industrial index (KBW bank index)5 and exchange rate.

From the Comparison of realized correlation of index and realized correlation of a specific

stock (DBK here), as expected, they follow the similar trends over time and the correlation of

index is more stable.

V. CONCLUSION

The exchange market has various forms and is among one of the most volatile and

unpredictable market in the worlds. In addition the unpredicted currency depreciation could

cause serious impact on the whole financial market. For instance, the Brazilian Real lost its value

up to 10% in the recent currency crisis in South America from early 2014. We can find the way

to predict the potential increasing volatility of exchange rate from the implied volatility

5 See Appendix 3 for definition and explanation for KBW Index

-0.5000

-0.4000

-0.3000

-0.2000

-0.1000

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

1

10 19 28 37 46 55 64 73 82 91

10

0

10

9

11

8

12

7

13

6

14

5

15

4

16

3


Realized Correlation (Index)


discrepancy of ADR options in U.S. market and home market. After the financial crisis in 2008,

the U.S. stock market yield a very low return and investors turned to foreign market especially

for emerging market like Brazil for higher return rate. Thus the portion of ADRs increased

significantly and the options market in such countries has rapidly developed, which gave us the

more adequate and statistically significant data. In the project, based on the result of summer

research collaborated with Professor Tim Leung and Connie Lee, I do more sensitivity analysis

for the implied exchange rate volatility respect to correlation between foreign stock market and

exchange rate. By presetting the correlation, the expression for implied exchange rate volatility

becomes simpler and explicit. There are three different measurements of correlation, short term

dynamic correlation, long-term correlation as regression and correlation with industrialized index.

In practice we can combine the measurements and get the more comprehensive impression about

correlation and prediction of implied exchange rate volatility.

The connection between ADR options volatility discrepancy and exchange rate can be further

explored. In the project we choose the bank industrial as example since the banks are more

sensitive to the exchange rate. Since the implied volatility of exchange rate is industry

independent we can possibly examine from other industrial with different sensitivity. All in all,

the methods connected the ADR market and exchange market and provided a good insight of

exchange market dynamics. It’s possible to make further and deeper exploration in the future.

REFERENCE:

[Tomas Bjork, 1998] Arbitrage Theory in Continuous Time chapter 12 Currency Derivatives

p167-181

[Stefan Eichler, Dominik Maltritz, 2008] Currency Crisis Prediction Using ADR Market Data -

An Option-Based Approach, International Journal of Forecasting 26

[Stefan Eichler, Alexander Karmann, Dominik Maltritz, 2009] The ADR Shadow Exchange

Rate as an Early Warning Indicator for Currency Crisis, Journal of Banking and Finance 33

[Dilip B. Madan, 2012] Joint Modeling the Prices of American Depository Receipts, the Local

Stock and the U.S. Dolalr, Journal of Investment Strategy Volume1/Number 4

[Louis Ederington, Wei Guan, 2005] The Information Frown in Option Prices, Journal of

Banking and Finance 29


Appendix 1

Name Country Stock

symbol

in US

Stock

symbol

in

original

country

Ratio

DR:ORD

Foreign

currency

Industry

UBS Switzerland UBS UBS 1:1 CHF Bank

Barclay British BCS BARC 1:4 GBX Bank

Santander Spain SAN SAN 1:1 EUR Bank

GlaxoSmithKline British GSK GSK 1:2 GBX healthcare

Novartis Switzerland NVS NOVN 1:1 CHF healthcare

TEVA Israel TEVA TEVA 1:1 NIS healthcare

British

Petroleum

British BP BP 1:6 GBX Oil and gas

Royal Dutch

Shell

British RDS RDS 1:2 GBX Oil and gas

Alcatel-Lucent France ALU ALU 1:1 EUR telecommunication

America Movil Mexico AML AML 1:20 MXN telecommunication

China Mobile China CHL 941 1:5 HKD telecommunication

Cemex Mexico CM CEMEX 1:10 MXN Building matrials

BHP Biliton Australia BHP BHP 1:2 AUD Metal and mining

Nokia Finland NOK NOK1V 1:1 EUR Tech equipment


Appendix 2

In the plots, the ADR volatility are shown in black and the original market volatility are shown in

red. All the strike price is converted into USD using the spot exchange rate assuming the

exchange rate doesn’t change until expiry. The blue doc points out the spot price and the blue

dashed lines are 75% and 125% of spot rate. Basically we only need compare the implied

volatility between the dashed lines.




Appendix 3

The KBW bank index is an economic index consisting of the stocks of 24 banking companies.

This index serves as a benchmark of the banking sector. This index trades on the Philadelphia

Stock Exchange, where it was created. The KBW Index is named after Keefe, Bruyette and

Woods, a recognized authority in the banking industry. The KBW Index trades under ticker

symbol BKX. The index is weighted according to capitalization and represents major banks and

money centers from across the country. Mathematically, the index is based on a tenth of the

value of the Keefe, Bruyette and Woods Index (KBWI). It began trading options in September of

1992.

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Joint Model for Exchange Rate Dynamics and Influence of ...aldous/Research/Ugrad/Jian_Wang.pdf · Influence of Presetting Correlation between Stock Price and Exchange Market ... dynamic